f(x)={[2^(x)," for "0

Question

f(x)={[2^(x)," for "0 < x <= 4],[8sqrtx," for "x > 4]:} Find  lim_(x rarr4)f(x). Choose 1 answer: (A) 4 (B) 8 (C) 16 (D) The limit doesn't exist.

Bytelearn · Accepted Answer

Find the limit of f(x) as x approaches 4: We need to find the limit of the function f(x) as x approaches 4. The function is defined differently for two intervals: when 0 < x <= 4, f(x) = 2^x, and when x > 4, f(x) = 8√x. To find the limit as x approaches 4, we need to consider the value of the function as x gets closer to 4 from both sides.Calculate the limit from the left side: First, let's consider the limit from the left side as x approaches 4. For x in the interval (0, 4], the function is defined as f(x) = 2^x. So we need to calculate the limit of 2^x as x approaches 4 from the left.Calculate the value of 2^4: Calculating the limit from the left, we have: lim_(x -> 4^-) 2^x = 2^4 This is because the function 2^x is continuous, and we can simply substitute the value of x with 4.Calculate the limit from the right side: Now, let's calculate the value of 2^4: 2^4 = 2 * 2 * 2 * 2 = 16 So, the limit from the left as x approaches 4 is 16.Consider the values of f(x) as x approaches 4: Next, we need to consider the limit from the right side as x approaches 4. For x in the interval (4, ∞), the function is defined as f(x) = 8√x. However, since we are only interested in the limit as x approaches 4, we do not need to evaluate this expression because it is not defined at x = 4.Check if the limits from the left and the right are equal: Since the function f(x) is not defined for x exactly equal to 4, we only consider the values of f(x) as x approaches 4 from the left and the right. We have already found that the limit from the left is 16. The limit from the right does not exist because the function changes at x = 4.Conclusion: The limit of f(x) as x approaches 4 is 16: To determine if the overall limit exists, we need to check if the limits from the left and the right are equal. Since the function changes at x = 4 and we only have the limit from the left, we conclude that the limit of f(x) as x approaches 4 is 16.

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